Let $f ( x ) = a _ { 0 } + a _ { 1 } | x | + a _ { 2 } | x | ^ { 2 } + a _ { 3 } | x | ^ { 3 }$, where $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$ are constants. Then (a) $f ( x )$ is differentiable at $x = 0$ whatever be $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$ (b) $f ( x )$ is not differentiable at $x = 0$ whatever be $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$ (c) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0$ (d) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0 , a _ { 3 } = 0$.
(c) Amongst $1 , | x | , | x | ^ { 2 } , | x | ^ { 3 }$, only $| x |$ is not differentiable at 0.
Let $f ( x ) = a _ { 0 } + a _ { 1 } | x | + a _ { 2 } | x | ^ { 2 } + a _ { 3 } | x | ^ { 3 }$, where $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$ are constants. Then\\
(a) $f ( x )$ is differentiable at $x = 0$ whatever be $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$\\
(b) $f ( x )$ is not differentiable at $x = 0$ whatever be $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$\\
(c) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0$\\
(d) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0 , a _ { 3 } = 0$.